Going to Mikvah on Shabbos:
A person is allowed to purify himself of his impurities by immersing in a mikvah on Shabbos even if this immersing is Biblically required [to be done], such as a nidda and the like.
The reason that this is allowed is: being that doing so does not appear like one is rectifying [something on Shabbos which is forbidden] but rather like one who is going [in the water in order] to cool himself off.
Dipping in reeky water, and dipping in the winter: [Furthermore] even to immerse in reeky water which is not commonly used to cool off in, and even [to immerse] in the winter when it is not at all common to [immerse in water to] cool oneself off, nevertheless [it is allowed as] at times when a person is dirty from mud and feces he washes himself even in the winter and even in reeky water in order to remove the mud and feces that are on him.
The custom regarding a woman’s immersion: [However] in our provinces the custom is to forbid the immersing of a woman on Shabbos unless her husband is in the city as well as that it was not possible for her to immerse before Shabbos or [she was able to but] her husband was not in the city and only arrived on Erev Shabbos in which case she was not negligent in not having immersed prior to Shabbos [See footnote for other Opinions]. However in any case that her husband was in the city and she was able to immerse [before Shabbos] and did not immerse then she may not immerse on Shabbos.
The reason behind this custom: There are authorities which have given [the following] rational behind this custom [saying] that since the custom has spread to not bathe on Shabbos therefore when an impure person immerses to purify himself it appears as if he is rectifying [an item which is forbidden to do on Shabbos] and not like he is going in [the water] to cool off, being that it is not at all common to bath in order to cool off due to the reason explained [above in Halacha 2] even though that there is no prohibition in doing so.
The custom regarding immersing for purification of nocturnal emission: Nevertheless a man is allowed to immerse [to purify himself] from a seminal discharge. [See Q&A regarding immersing for prayer]
The reason is because: since this immersion is not Biblically required (and is not even a complete Rabbinical obligation [therefore] it does not appear like one is rectifying [himself].)
The immersion of a penitent: (As well an apostate that repented is allowed to immerse on Shabbos because this immersion is not Biblically required as opposed to the immersion of a convert (as well as that there is not even a complete Rabbinical obligation [for the penitent to do so])).
A woman who needs to re-immerse on Shabbos due to a stringency: As well any woman which needs to re-immerse due to a mere stringency is permitted to immerse on Shabbos as explained in Yorah Deah chapter 197 [Halacha 1 in Shach].
Summary-Going to Mikvah on Shabbos:
For penitence or for Keri: Is permitted to be done in cold water, even if the water is murky, and even in the winter.
A Nidah: Is from the letter of the law permitted to immerse, although the custom is to only allow her to immerse if her husband is in town and she did not have a chance to immerse prior to Shabbos, or had a chance but her husband only arrived on Erev Shabbos. A woman may always immerse if she is only required to do so as a mere stringency. [According to all a women may immerse on Shabbos if she was unable to do so due to a Physical impalement, such as being after birth.]
Q&A on a women’s Immersion [Yoreh Deah 197]
What is the law if one transgressed and immersed in a case that she was not allowed to do so, is she considered pure?
If this was done advertently, despite knowledge of the prohibition, then she remains prohibited to her husband until after Shabbos.
If however this was done inadvertently, without prior knowledge of the prohibition, then she is permitted to her husband.
Does the above prohibition apply also on Yom Tov? 
Some opinions say that Yom Tov initially maintains the same laws as do Shabbos in this regard, however stating that Bedieved, after the fact, she is considered pure even if she immersed advertently despite knowing of the prohibition.
Other Poskim however argue on this saying that there is no prohibition at all to immerse on Yom Tov in any circumstance and she may do so even initially.
If one was not required to wait 5 days in those cases in which it is explained that there is no need to do so, but did so anyways, may she immerse on Shabbos if that’s when her 7th clean day falls on?
Some Poskim rule this is forbidden to be done, and if one did so she may not immerse on Friday night. Others however rule it is permitted to be done and she may therefore immerse on Friday night.
If by mistake a woman did not do her Hefsek Taharah on time [before Sunset], and this thus caused her Mikvah night to be pushed off a day later which is Friday night, may she immerse?
May a woman after birth push off her Hefsek Taharah to Friday which may cause her to go to Mikveh on Shabbos?
She may do so according to all if she does not feel physically ready to go to Mikveh until that time. If she does feel ready beforehand and pushed it off for other reasons, then this matter is disputed as mentioned above.
Q&A on Men immersing in Mikveh on Shabbos
According to today’s custom to not bathe at all on Shabbos may one nevertheless immerse in a Mikveh on Shabbos?
Yes, not only is it allowed but it is even a Mitzvah to immerse for the sake of purification from nocturnal emission or even for additional purity as is customarily done on Shabbos morning prior to prayer. However one must nevertheless take caution not to squeeze any of his hair. [See footnote for opinion of M”B]
How many times is one to immerse in the Mikveh on Shabbos day?
It suffices to immerse one time. [Although if one is a Baal Keri] he is to immerse twice.
May one who desires to immerse in a Mikveh do so even in a river or sea?
One who needs to immerse due to nocturnal emission may do so even in a river or ocean, although taking care to avoid transgressing any of the matters mentioned in Halacha 2. [This opinion seems to be the opinion of Admur, and so rules Ketzos Hashulchan.]
However there are Poskim which prohibit immersing in any water that is not within a private domain.
May one immerse in a hot Mikveh on Shabbos?
Below Yad Soledes [110 Fahrenheit]: Is defined as warm according to Admur [see Halacha 1D] and is thus allowed without question. However according to those Poskim which define warm as “no longer called hot” or below 98.6 Fahrenheit, then it must be below this temperature to be allowed without question, otherwise it is the same as water above Yad Soledes.
Above Yad Soledes: Some Poskim prohibit even women from immersing in Yad Soledes water, based on the prohibition against bathing the majority of one’s body in hot water on Shabbos [See Halacha 1]
Other Poskim however permit women to immerse in even hot water and they write that so is the custom.
Nevertheless one may not remain in the water for pure pleasure purposes and is rather to immerse and leave. Those that stay in the water are to be protested against doing so [if it is above 110].
If the water was heated on Shabbos through a timer or gentile? Some are accustomed to be lenient to allow immersing in such water however many have written against doing so. The Igros Moshe allows one to Bedieved use such a Mikveh even if it was heated by a Jew on Shabbos. [Opinion of Admur: Seemingly according to the ruling of Admur above in Halacha 1 that even water warmed on Shabbos in a permitted way may not be placed on one’s body, this would be forbidden.]
May one swim in a Mikveh on Shabbos?
Depends on the Mikveh.
May one splash away the dirt that floats on the surface of a Mikvah?
May one remove the stopper which attaches the Mikveh to the pit of rain water?
 So rules Michaber 326/8
 See Rama Yoreh Deah 197/2 which implies that it is forbidden from the letter of the law. However to note that in Orach Chayim 326/8 the Rama does not make any mention against bathing in a river or against immersing on Shabbos, of which the Michaber rules there that it is permitted. Vetzaruch Iyun as to why no gloss was written by the Rama being that he in truth holds that it is forbidden, as he writes in Yoreh Deah. Perhaps then one can say that the Rama too agrees that it is permitted from the letter of the law and that just due to the custom it is forbidden, as is the common explanation in other contradictory rulings. Vetzaruch Iyun.
 This follows the lenient opinion in Yoreh Deah 197/2. The Rama there brings the stringent opinion and rules that “so is the custom in certain places, however in a community where there is no custom then one is not to be stringent.” As rules Rama, so rules also the Bach and Shach [Shach 197/3, based on their understanding of the Terumas Hadeshen] that the main opinion is like the lenient opinion and thus in a place where there is no custom one may be lenient.
However Admur rules simply like the lenient opinion that she may immerse in such a case and does not make any mention of customs.
The Beis Yosef [brought in Shach 197/3] rules that there is never a prohibition to immerse on Shabbos in any situation. [Vetzaruch Iyun if this is even if she could have immersed before hand? Seemingly yes, as so is implied from the fact that the Michaber never rules anywhere regarding any restrictions in immersing on Shabbos, and rather wrote simply that one may immerse on Shabbos.]
However the Taz [197/4] rules [based on his understanding of the Terumas Hadeshen] like the stringent opinion in Rama that it is forbidden to immerse on Shabbos even if [she had holy reasons to avoid doing so before hand, such as that] her husband did not arrive until Erev Shabbos. However he too agrees that in a case that she could not immerse due to a physical impracticality [Oness], such as that she is after birth and did not have the strength to immerse beforehand, then it is allowed. Thus the Taz argues on Rama which holds that it is dependent on the custom and rather rules that it is always forbidden unless it was physically impossible for her to do so. In the Nekudos Hakesef the Shach debates against the ruling of the Taz and supports the ruling of the Rama and Bach.
 Taz – Yoreh Deah 197/4
 However the Gr”a, brought in Biur Halacha rules that since today we are accustomed not to bathe even in cold water it therefore appears like one is fixing and one may thus not immerse for nocturnal emission. The Ketzos Hashulchan 133 footnote 8 however adds that today since many are accustomed to immerse for the sake of additional purity [which does not involve a fixing prohibition according to all] it therefore once again is unnoticeable that one is immersing for nocturnal emission and thus even according to the GR”A it would be allowed.
 Taz 197/4
 Toras Hashelmim 197/3/Pischeiy Teshuvah 5 in name of Shut Chut Hashani
 Toras Hashelamim 197/3
 Shut Chut Hashani
 Toras Hashelamim 197/3
 Nodeh Beyehuda 131; Pischeiy Teshuvah 197/4
 Daas Kedoshim in Daas Torah Mahrsham; Avnei Nezer Yoreh Deah 246
 Nodeh Beyehudah, brought in Pischeiy Teshuvah 197/2
 Ketzos Hashulchan 133/4 footnote 8
 The reason for this is because in an indoor Mikveh one need only be careful regarding squeezing the hair, which is easily avoidable, while the carrying and swimming issues are not relevant. Thus one may be more lenient.
Now although Admur wrote that “the custom is to not bathe at all on Shabbos “, nevertheless it is evident from the fact that later on Admur states the permission to immerse that this custom only applies to pleasure baths and not to baths motivated for other purposes. [ibid]
 As writes the Arizal and other Mekubalim that one is to immerse on Shabbos. [ibid] The Arizal writes that if one is a Baal Keri or had Tashmish on Friday night then he is obligated to immerse on Shabbos day. Furthermore, he writes, even if the above did not occur one must still immerse on Shabbos day in order to garb his soul with the new revelations of Shabbos day, which one did not yet receive with his immersion done on Erev Shabbos. [Kaf Hachayim 260/6]
 So is the custom of majority of the Torah sages in Jerusalem of all sects, including the Sefaradim and Perushim. [ibid]
 The M”B however concludes in Biur Halacha that one is to avoid immersing for purposes of additional purity being that one may come to squeeze his hair, and it suffices that we already allow to be lenient to immerse for nocturnal emission. The Ketzos Hashulchan 133 footnote 8 however takes to par this ruling of the M”B being that a)The source of the M”B is in the Mahril which sates with regards to women that would avoid immersing on Shabbos if that is not their Halachic date of immersing. Now the Mahril states that the reason for this stringency is to avoid any carrying of the water, swimming, squeezing hair. Now, regarding men immersing in an indoor Mikveh only the squeezing of hair is relevant and thus there is no source from the Mahril to be stringent when only one suspicion is relevant. Furthermore the suspicion of squeezing hair is a lot more relevant by women than men, and thus there is no support from the ruling regarding women at all.
b) The Arizal writes that it is a Mitzvah to immerse on Shabbos morning and it is thus considered its right time, as opposed to the ruling by women which involves a date which is not their correct time.
c) The custom of not bathing in cold waters is itself a stringency and it thus suffices to be stringent to not bathe for pleasure, and not to extend this to even Mitzvah purposes. [ibid]
 Kaf Hachayim 260/6
 So writes Arizal, in order to garb his soul with the holiness of Shabbos day. [ibid]
 Ketzos Hashulchan 133 footnote 8
 So rules Tosefes Shabbos and so is implied from Admur here which does not differentiate between bodies of water.
 Minchas Yitzchak 6/32, Piskeiy Teshuvos 339/1
 Regarding the Kashrus of a Hot Mikveh from the aspect of the laws of Mikvaos see Yoreh Deah 201/75 that Michaber forbids this even during the week [due to people coming to think that a bathhouse is also a Kosher Mikveh-Taz] while the Rama rules that in those places that are lenient may do so, although one is to be stringent . See as well Tikunei Mikvaos where Admur writes that hot water is placed into the Mikveh.
 See Pischeiy Teshuvah Yorah Deah 197/1 for a compilation of different opinions.
 Chacham Tzevi 11, Nodah Beyehudah Tenyana Orach Chayim 24 [so writes Pischeiy Teshuvah as well as Divrei Chayim, however Tzitz Eliezer argues on their opinion in Nodah Beyehudah.] The Sheivet Halevy [5/44] rules that it is best to immerse in a warm Mikveh if he is able to, despite the fact that the custom is to be lenient.
 Karban Nesanel [Perek Bameh Madlikin 22/11] says that this is not included in the decree against bathing; Bircheiy Yosef Yoreh Deah 197/2; Divrei Yosef, although he concludes with a Tzaruch Iyun [so says Pischeiy Teshuvah, however Tzitz Eliezer differs that he does allow it and so learns others as well], Tzitz Eliezer 6/22 which brings many opinions which permit it; Divrei Chayim 2/26, Avnei Nezer 1/526. The Sheivet Halevy [5/44] rules that it is best to immerse in a warm Mikveh if he is able to, despite the fact that the custom is to be lenient.
 So rules Avnei Nezer 1/526
 Piskeiy Teshuvos 326/2
 Vetzaruch Iyun as perhaps since it’s not being done for purposes of bathing it is permitted even if the water is heated on Shabbos, as from the aspect of bathing the entire body it is equally forbidden to be done whether with water heated before or on Shabbos.
 Ketzos Hashulchan 146 footnote 34/17 and 35; Shabbos Kihalacha Vol. 2 p. 242.
Splashing away dirt may involve the “Swimming” prohibition as well as the Borer prohibition .
 Piskeiy Teshuvos 226/5
 So rules Darcheiy Teshuvah Yoreh Deah 201 which brings many opinions which allow this as to opposed to the Mahrshik which writes that its forbidden.
So rules also that it is allowed: Igros Moshe [did not find source], Dvar Moshe, Mishneh Halachos.
 As purifying impure waters is allowed to be done on Shabbos through pouring them into a Mikveh. [Rambam Shabbos 23/8]
 Removing a pit cover may contain two prohibitions: Muktzah if it does not have a handle [308/37] and Setirah [if it is not made to constantly remove]. In this case neither prohibition applies as it is made to constantly open and close, as well as that it appears that it is made for doing so despite the fact that it does not contain a handle. [Sheivet Hakehasi 3/132]